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The Monomorphism s in Ab are the Injective group homomorphisms, the Epimorphism s are the Surjective group homomorphisms, and the Isomorphism s are the Bijective group homomorphisms. The Zero Object of Ab is the trivial group {0} which consists only of its Neutral Element . Note that Ab is a Full Subcategory of '''Grp''', the Category Of ''all'' Groups . The main difference between Ab and '''Grp''' is that the sum of two homomorphisms ''f'' and ''g'' between abelian groups is again a group homomorphism: :(''f''+''g'')(''x''+''y'') = ''f''(''x''+''y'') + ''g''(''x''+''y'') = ''f''(''x'') + ''f''(''y'') + ''g''(''x'') + ''g''(''y'') : = ''f''(''x'') + ''g''(''x'') + ''f''(''y'') + ''g''(''y'') = (''f''+''g'')(''x'') + (''f''+''g'')(''y'') The third equality requires the group to be abelian. This addition of morphism turns Ab into a Preadditive Category , and because the Direct Sum of finitely many abelian groups yields a Biproduct , we indeed have an Additive Category . In Ab, the notion of of ''B'', and that therefore the quotient group ''B''/''f''(''A'') cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that Ab is indeed an Abelian Category . The Product in Ab is given by the Product Of Groups , formed by taking the Cartesian Product of the underlying sets and performing the group operation componentwise. Because Ab has kernels, one can then show that Ab is a Complete Category . The Coproduct in Ab is given by the Direct Sum of groups; since Ab has cokernels, it follows that Ab is also Cocomplete . Taking Direct Limit s in Ab is an Exact Functor , which turns Ab into an Ab5 Category . We have a Forgetful Functor Ab → '''Set''' which assigns to each abelian group the underlying Set , and to each group homomorphism the underlying Function . This functor is Faithful , and therefore Ab is a Concrete Category . The forgetful functor has a Left Adjoint (which associates to a given set the Free Abelian Group with that set as basis) but does not have a right adjoint. An object in Ab is Injective if and only if it is Divisible ; it is Projective if and only if it is a free abelian group. The category has a projective generator ('''Z''') and an Injective Cogenerator ('''Q'''/'''Z'''). Given two abelian groups ''A'' and ''B'', their Tensor Product ''A''⊗''B'' is defined; it is again an abelian group. With this notion of product, Ab is a Symmetric Strict Monoidal Category . Ab is not Cartesian Closed (and therefore also not a Topos ) since it lacks Exponential Object s. |
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